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68
THE HIDDEN SUBGROUP PROBLEM  REVIEW AND OPEN PROBLEMS
, 2004
"... An overview of quantum computing and in particular the Hidden Subgroup Problem are presented from a mathematical viewpoint. Detailed proofs are supplied for many important results from the literature, and notation is unified, making it easier to absorb the background necessary to begin research on ..."
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Cited by 19 (1 self)
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An overview of quantum computing and in particular the Hidden Subgroup Problem are presented from a mathematical viewpoint. Detailed proofs are supplied for many important results from the literature, and notation is unified, making it easier to absorb the background necessary to begin research on the Hidden Subgroup Problem. Proofs are provided which give very concrete algorithms and bounds for the finite abelian case with little outside references, and future directions are provided for the nonabelian case. This summary is current as of October 2004.
Shor’s algorithm on a nearestneighbor machine
 Asian conference on Quantum Information Science
, 2007
"... We give a new “nested adds ” circuit for implementing Shor’s algorithm in linear width and quadratic depth on a nearestneighbor machine. Our circuit combines Draper’s transform adder with approximation ideas of Zalka. The transform adder requires small controlled rotations. We also give another ver ..."
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Cited by 17 (1 self)
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We give a new “nested adds ” circuit for implementing Shor’s algorithm in linear width and quadratic depth on a nearestneighbor machine. Our circuit combines Draper’s transform adder with approximation ideas of Zalka. The transform adder requires small controlled rotations. We also give another version, with slightly larger depth, using only reversible classical gates. We do not know which version will ultimately be cheaper to implement. 1
The Quantum Fourier Transform and Extensions of the Abelian Hidden Subgroup Problem
, 2002
"... The quantum Fourier transform (QFT) has emerged as the primary tool in quantum algorithms which achieve exponential advantage over classical computation and lies at the heart of the solution to the abelian hidden subgroup problem, of which Shor’s celebrated factoring and discrete log algorithms are ..."
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Cited by 12 (0 self)
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The quantum Fourier transform (QFT) has emerged as the primary tool in quantum algorithms which achieve exponential advantage over classical computation and lies at the heart of the solution to the abelian hidden subgroup problem, of which Shor’s celebrated factoring and discrete log algorithms are a special case. We begin by addressing various computational issues surrounding the QFT and give improved parallel circuits for both the QFT over a power of 2 and the QFT over an arbitrary cyclic group. These circuits are based on new insight into the relationship between the discrete Fourier transform over different cyclic groups. We then exploit this insight to extend the class of hidden subgroup problems with efficient quantum solutions. First we relax the condition that the underlying hidden subgroup function be distinct on distinct cosets of the subgroup in question and show that this relaxation can be solved whenever G is a finitelygenerated abelian group. We then extend this reasoning to the hidden cyclic subgroup problem over the reals, showing how to efficiently generate the bits of the period of any sufficiently piecewisecontinuous function on ℜ. Finally, we show that this problem of periodfinding over ℜ, viewed as an oracle promise problem, is strictly harder than its integral counterpart. In particular, periodfinding
New Limits on FaultTolerant Quantum Computation
, 2008
"... We show that quantum circuits cannot be made faulttolerant against a depolarizing noise level of ˆ θ = (6 − 2 √ 2)/7 ≈ 45%, thereby improving on a previous bound of 50 % (due to Razborov [18]). Our precise quantum circuit model enables perfect gates from the Clifford group (CNOT, Hadamard, S, X, Y, ..."
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Cited by 12 (2 self)
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We show that quantum circuits cannot be made faulttolerant against a depolarizing noise level of ˆ θ = (6 − 2 √ 2)/7 ≈ 45%, thereby improving on a previous bound of 50 % (due to Razborov [18]). Our precise quantum circuit model enables perfect gates from the Clifford group (CNOT, Hadamard, S, X, Y, Z) and arbitrary additional onequbit gates that are subject to depolarizing noise ˆ θ. We prove that this set of gates cannot be universal for arbitrary (even classical) computation, from which the upper bound on the noise threshold for faulttolerant quantum computation follows.
Quantum Lower Bounds for Fanout
, 2003
"... We prove several new lower bounds for constant depth quantum circuits. The main result is that parity (and hence fanout) requires log depth circuits, when the circuits are composed of single qubit and arbitrary size Tooli gates, and when they use only constantly many ancill. Under this constraint, t ..."
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Cited by 9 (3 self)
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We prove several new lower bounds for constant depth quantum circuits. The main result is that parity (and hence fanout) requires log depth circuits, when the circuits are composed of single qubit and arbitrary size Tooli gates, and when they use only constantly many ancill. Under this constraint, this bound is close to optimal. In the case of a nonconstant number a of ancill and n input qubits, we give a tradeo between a and the required depth, that results in a nontrivial lower bound for fanout when a = n 1 o(1) .
Quantum fanout is powerful
 Theory of Computing
"... We demonstrate that the unbounded fanout gate is very powerful. Constantdepth polynomialsize quantum circuits with bounded fanin and unbounded fanout over a fixed basis (denoted by QNC 0 f) can approximate with polynomially small error the following gates: parity, mod[q], And, Or, majority, thr ..."
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We demonstrate that the unbounded fanout gate is very powerful. Constantdepth polynomialsize quantum circuits with bounded fanin and unbounded fanout over a fixed basis (denoted by QNC 0 f) can approximate with polynomially small error the following gates: parity, mod[q], And, Or, majority, threshold[t], exact[t], and Counting. Classically, we need logarithmic depth even if we can use unbounded fanin gates. If we allow arbitrary onequbit gates instead of a fixed basis, then these circuits can also be made exact in logstar depth. Sorting, arithmetic operations, phase estimation, and the quantum Fourier transform with arbitrary moduli can also be approximated in constant depth. 1
On the Design and Optimization of a Quantum PolynomialTime Attack on Elliptic Curve Cryptography
, 710
"... Abstract. We consider a quantum polynomialtime algorithm which solves the discrete logarithm problem for points on elliptic curves over GF(2 m). We improve over earlier algorithms by constructing an efficient circuit for multiplying elements of binary finite fields and by representing elliptic curv ..."
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Abstract. We consider a quantum polynomialtime algorithm which solves the discrete logarithm problem for points on elliptic curves over GF(2 m). We improve over earlier algorithms by constructing an efficient circuit for multiplying elements of binary finite fields and by representing elliptic curve points using a technique based on projective coordinates. The depth of our proposed implementation is O(m 2), which is an improvement over the previous bound of O(m 3). 1
On the effect of quantum interaction distance on quantum addition circuits
 J. Emerg. Technol. Comput. Syst
, 2011
"... ar ..."